Newspace parameters
comment: Compute space of new eigenforms
[N,k,chi] = [91,10,Mod(5,91)]
mf = mfinit([N,k,chi],0)
lf = mfeigenbasis(mf)
from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(91, base_ring=CyclotomicField(12))
chi = DirichletCharacter(H, H._module([10, 9]))
N = Newforms(chi, 10, names="a")
//Please install CHIMP (https://github.com/edgarcosta/CHIMP) if you want to run this code
chi := DirichletCharacter("91.5");
S:= CuspForms(chi, 10);
N := Newforms(S);
Level: | \( N \) | \(=\) | \( 91 = 7 \cdot 13 \) |
Weight: | \( k \) | \(=\) | \( 10 \) |
Character orbit: | \([\chi]\) | \(=\) | 91.bb (of order \(12\), degree \(4\), minimal) |
Newform invariants
comment: select newform
sage: f = N[0] # Warning: the index may be different
gp: f = lf[1] \\ Warning: the index may be different
Self dual: | no |
Analytic conductor: | \(46.8682610909\) |
Analytic rank: | \(0\) |
Dimension: | \(328\) |
Relative dimension: | \(82\) over \(\Q(\zeta_{12})\) |
Twist minimal: | yes |
Sato-Tate group: | $\mathrm{SU}(2)[C_{12}]$ |
$q$-expansion
The dimension is sufficiently large that we do not compute an algebraic \(q\)-expansion, but we have computed the trace expansion.
Embeddings
For each embedding \(\iota_m\) of the coefficient field, the values \(\iota_m(a_n)\) are shown below.
For more information on an embedded modular form you can click on its label.
comment: embeddings in the coefficient field
gp: mfembed(f)
Label | \( a_{2} \) | \( a_{3} \) | \( a_{4} \) | \( a_{5} \) | \( a_{6} \) | \( a_{7} \) | \( a_{8} \) | \( a_{9} \) | \( a_{10} \) | ||||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|
5.1 | −11.4807 | − | 42.8465i | −49.6096 | + | 28.6421i | −1260.61 | + | 727.814i | −39.7613 | + | 10.6540i | 1796.77 | + | 1796.77i | −6247.28 | + | 1151.15i | 29597.7 | + | 29597.7i | −8200.76 | + | 14204.1i | 912.974 | + | 1581.32i |
5.2 | −10.9276 | − | 40.7824i | 166.664 | − | 96.2237i | −1100.38 | + | 635.307i | 202.056 | − | 54.1409i | −5745.47 | − | 5745.47i | 1613.04 | − | 6144.24i | 22648.2 | + | 22648.2i | 8676.50 | − | 15028.1i | −4415.99 | − | 7648.71i |
5.3 | −10.8870 | − | 40.6310i | 120.266 | − | 69.4354i | −1088.94 | + | 628.702i | −348.043 | + | 93.2579i | −4130.57 | − | 4130.57i | 4183.56 | + | 4780.32i | 22171.2 | + | 22171.2i | −198.943 | + | 344.579i | 7578.32 | + | 13126.0i |
5.4 | −10.8713 | − | 40.5723i | −111.169 | + | 64.1836i | −1084.52 | + | 626.147i | −1911.08 | + | 512.072i | 3812.63 | + | 3812.63i | 3916.23 | − | 5001.68i | 21987.4 | + | 21987.4i | −1602.44 | + | 2775.51i | 41551.8 | + | 71969.9i |
5.5 | −10.6545 | − | 39.7631i | −221.572 | + | 127.925i | −1024.18 | + | 591.312i | 1246.07 | − | 333.883i | 7447.41 | + | 7447.41i | 4766.00 | + | 4199.86i | 19520.9 | + | 19520.9i | 22887.9 | − | 39642.9i | −26552.5 | − | 45990.2i |
5.6 | −10.1912 | − | 38.0341i | −54.7649 | + | 31.6185i | −899.330 | + | 519.228i | −1311.11 | + | 351.311i | 1760.70 | + | 1760.70i | 4438.33 | + | 4544.76i | 14658.1 | + | 14658.1i | −7842.04 | + | 13582.8i | 26723.6 | + | 46286.7i |
5.7 | −10.1645 | − | 37.9344i | 9.27740 | − | 5.35631i | −892.300 | + | 515.170i | 2307.23 | − | 618.220i | −297.489 | − | 297.489i | 1373.49 | − | 6202.19i | 14394.3 | + | 14394.3i | −9784.12 | + | 16946.6i | −46903.7 | − | 81239.5i |
5.8 | −10.1089 | − | 37.7268i | −116.736 | + | 67.3977i | −877.720 | + | 506.752i | 1557.93 | − | 417.446i | 3722.77 | + | 3722.77i | −5979.50 | + | 2144.56i | 13850.5 | + | 13850.5i | −756.597 | + | 1310.46i | −31497.9 | − | 54555.9i |
5.9 | −9.88529 | − | 36.8924i | 230.291 | − | 132.958i | −819.927 | + | 473.385i | −1552.67 | + | 416.037i | −7181.65 | − | 7181.65i | −5367.23 | + | 3398.00i | 11741.9 | + | 11741.9i | 25514.4 | − | 44192.2i | 30697.2 | + | 53169.1i |
5.10 | −9.84620 | − | 36.7465i | 121.299 | − | 70.0318i | −809.953 | + | 467.627i | 1878.59 | − | 503.366i | −3767.76 | − | 3767.76i | −468.182 | + | 6335.17i | 11385.7 | + | 11385.7i | −32.5834 | + | 56.4360i | −36993.9 | − | 64075.2i |
5.11 | −9.41397 | − | 35.1334i | 55.9682 | − | 32.3132i | −702.328 | + | 405.489i | −1552.01 | + | 415.860i | −1662.16 | − | 1662.16i | −5033.95 | − | 3874.65i | 7689.56 | + | 7689.56i | −7753.21 | + | 13429.0i | 29221.2 | + | 50612.6i |
5.12 | −9.08088 | − | 33.8903i | −163.524 | + | 94.4107i | −622.686 | + | 359.508i | 197.881 | − | 53.0221i | 4684.55 | + | 4684.55i | 1215.08 | − | 6235.16i | 5135.93 | + | 5135.93i | 7985.28 | − | 13830.9i | −3593.87 | − | 6224.77i |
5.13 | −8.94964 | − | 33.4005i | −222.806 | + | 128.637i | −592.093 | + | 341.845i | −2577.17 | + | 690.551i | 6290.58 | + | 6290.58i | −4593.27 | + | 4388.10i | 4197.96 | + | 4197.96i | 23253.6 | − | 40276.4i | 46129.5 | + | 79898.7i |
5.14 | −8.43697 | − | 31.4872i | −61.0517 | + | 35.2482i | −476.857 | + | 275.313i | 1307.03 | − | 350.218i | 1624.96 | + | 1624.96i | 6319.13 | + | 649.753i | 890.360 | + | 890.360i | −7356.63 | + | 12742.1i | −22054.8 | − | 38200.0i |
5.15 | −7.84691 | − | 29.2851i | 148.521 | − | 85.7485i | −352.636 | + | 203.594i | 1538.27 | − | 412.177i | −3676.58 | − | 3676.58i | −6345.80 | + | 290.604i | −2246.96 | − | 2246.96i | 4864.12 | − | 8424.90i | −24141.2 | − | 41813.9i |
5.16 | −7.81263 | − | 29.1571i | −10.1416 | + | 5.85528i | −345.695 | + | 199.587i | −1154.98 | + | 309.476i | 249.956 | + | 249.956i | −2271.29 | + | 5932.53i | −2408.20 | − | 2408.20i | −9772.93 | + | 16927.2i | 18046.8 | + | 31258.0i |
5.17 | −7.64658 | − | 28.5374i | 145.433 | − | 83.9657i | −312.510 | + | 180.428i | −2483.48 | + | 665.446i | −3508.23 | − | 3508.23i | 6197.43 | − | 1394.80i | −3157.54 | − | 3157.54i | 4258.97 | − | 7376.76i | 37980.2 | + | 65783.7i |
5.18 | −7.42068 | − | 27.6944i | 70.6230 | − | 40.7742i | −268.507 | + | 155.022i | −828.537 | + | 222.006i | −1653.29 | − | 1653.29i | −4105.26 | − | 4847.72i | −4094.38 | − | 4094.38i | −6516.43 | + | 11286.8i | 12296.6 | + | 21298.4i |
5.19 | −7.37110 | − | 27.5093i | 45.1180 | − | 26.0489i | −259.024 | + | 149.548i | 378.159 | − | 101.327i | −1049.16 | − | 1049.16i | 6292.42 | − | 871.216i | −4287.52 | − | 4287.52i | −8484.41 | + | 14695.4i | −5574.89 | − | 9655.99i |
5.20 | −7.27837 | − | 27.1633i | −177.038 | + | 102.213i | −241.463 | + | 139.408i | 171.168 | − | 45.8642i | 4064.99 | + | 4064.99i | −5809.82 | − | 2568.98i | −4636.82 | − | 4636.82i | 11053.5 | − | 19145.3i | −2491.64 | − | 4315.65i |
See next 80 embeddings (of 328 total) |
Inner twists
Char | Parity | Ord | Mult | Type |
---|---|---|---|---|
1.a | even | 1 | 1 | trivial |
7.d | odd | 6 | 1 | inner |
13.d | odd | 4 | 1 | inner |
91.bb | even | 12 | 1 | inner |
Twists
By twisting character orbit | |||||||
---|---|---|---|---|---|---|---|
Char | Parity | Ord | Mult | Type | Twist | Min | Dim |
1.a | even | 1 | 1 | trivial | 91.10.bb.a | ✓ | 328 |
7.d | odd | 6 | 1 | inner | 91.10.bb.a | ✓ | 328 |
13.d | odd | 4 | 1 | inner | 91.10.bb.a | ✓ | 328 |
91.bb | even | 12 | 1 | inner | 91.10.bb.a | ✓ | 328 |
By twisted newform orbit | |||||||
---|---|---|---|---|---|---|---|
Twist | Min | Dim | Char | Parity | Ord | Mult | Type |
91.10.bb.a | ✓ | 328 | 1.a | even | 1 | 1 | trivial |
91.10.bb.a | ✓ | 328 | 7.d | odd | 6 | 1 | inner |
91.10.bb.a | ✓ | 328 | 13.d | odd | 4 | 1 | inner |
91.10.bb.a | ✓ | 328 | 91.bb | even | 12 | 1 | inner |
Hecke kernels
This newform subspace is the entire newspace \(S_{10}^{\mathrm{new}}(91, [\chi])\).